Back17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = 9−t² on [0, 4]; s(0)=−2
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = 6−2t on [0, 5]; s(0)=0
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = −t³+3t²−2t on [0, 3]; s(0)=4
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
b. Find the displacement over the given interval.
v(t) = 50e^−2t on [0, 4]
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
a. Determine when the motion is in the positive direction and when it is in the negative direction.
v(t) = 50e^−2t on [0, 4]
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
c. Find the distance traveled over the given interval.
v(t) = 4t³ - 24t²+20t on [0, 5]
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
c. Find the distance traveled over the given interval.
v(t) = 3t²−6t on [0, 3]
13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.
b. Find the displacement over the given interval.
v(t) = 3t²−6t on [0, 3]
Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.
c. Find the distance traveled by the object on the interval 0≤t≤6.
Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.
b. Find the displacement of the object on the interval 0≤t≤6.
Given the velocity function of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.
An oscillator The acceleration of an object moving along a line is given by a(t) = 2 sin πt/4. The initial velocity and position are v(0)= −8/π and s(0)=0.
a. Find the velocity and position for t≥0.
Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.
d. How long does it take the racer to travel 300 ft?
Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.
e. How far has the racer traveled when it reaches a speed of 178 ft/s?
Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.
b. How far does the car travel in the time it takes to come to rest?